MathDB

(183) 6

Part of 1988 Tournament Of Towns

Problems(1)

TOT 183 1988 Spring S6 not periodic 2 letter word sequence

Source:

5/20/2020
Consider a sequence of words , consisting of the letters AA and BB . The first word in the sequence is "AA" . The k-th word i s obtained from the (kāˆ’1)(k-1)-th by means of the following transformation : each AA is substituted by AABAAB , and each BB is substituted by AA. It is easily seen that every word is an initial part of the next word. The initial parts of these words coincide to give a sequence of letters AABAABAAABAABAAB...AABAABAAA BAABAAB... (a) In which place of this sequence is the 10001000-th letter AA? (b ) Prove that this sequence is not periodic.
(V . Galperin , Moscows)
periodicalWordscombinatorics